Vibration of a Helicopter Planetary Gear: Experiments

and Analytical Simulation

Tristan M. Ericson

Department of Mechanical and Aerospace Engineering

Ohio State University

Columbus, Ohio 43210

ericson.15@osu.edu

March 4, 2011

Abstract

Planetary gear vibration is a major source of noise and may lead to fatigue-induced fail-

ures in bearings or other drivetrain components. Gear designers use mathematical models to

analyze potential designs, but these models remain unverified by experimental data. This paper

presents experiments that completely characterize the dynamic behavior of a planetary gear by

modal testing and spinning tests under operating conditions, focusing on the independent mo-

tion of planetary components. Accelerometers are mounted directly to individual gear bodies.

Rotational and translational accelerations obtained from the experiments are compared to the

predictions of a lumped parameter model. Natural frequencies and modes agree well. Forced

response also agrees well between experiments and the model. Rotational, translational, and

planet mode types presented in published analytical research are observed experimentally.

1

INTRODUCTION

Planetary gears are used in many commercial and military applications: helicopters, automobiles,

wind turbines, agricultural equipment, and more. They usually transmit high loads, leading to

increased noise and vibration. Excessive vibration may cause failures at the gears or propagate

through bearings to the surrounding drivetrain components and housing. Several computer models

are presented in the research literature [1–6] that allow engineers to predict the dynamic behavior

of planetary gear designs. These models, however, are not verified by controlled experiments that

demonstrate the independent motion of planetary gear components.

Early planetary gear research focused on analytical models of spur gears, studying the relation-

ship between natural frequencies and system parameters. Cunliffe et al. [1] explored the character-

istics of vibration modes in a 13 degree-of-freedom planetary gear with a fixed carrier. Botman [2]

studied the modes of an 18 degree-of-freedom system and the effects of planet pin stiffness on

the natural frequencies. August and Kasuba [3] used a 9 degree-of-freedom model to compare the

performance of fixed versus floating sun gears and argue that a fixed sun gear system with accurate

gearing outperforms a floating sun system. Saada and Velex [4] presented a model for planetary

gears with spur or helical gears. Abousleiman et al. [7] later included flexibility of the ring gear

and carrier. Kahraman [5] proposed a simplified lumped parameter model, providing closed-form

expressions for torsional natural frequencies in terms of system parameters. Bahk and Parker used

analytical and computational methods to find nonlinear behavior in an equally spaced spur plan-

etary gear [8]. Assuming a symmetric system with even load sharing, Lin and Parker [6] showed

that a two-dimensional planetary system has unique modal properties. They classified all modes

into three types, each with distinct characteristics: rotational, translational, and planet modes.

Finite element methods are also used to study planetary gear dynamics. Abousleiman and

Velex [9] developed a hybrid three-dimensional finite element/lumped parameter model and used

it to analyze planetary gear dynamics with a flexible annulus and planet carrier [7]. Vijayakar [10]

developed a specialized finite element/contact mechanics program that permits a relatively coarse

mesh near the contact region to allow dynamic solution. Parker et al. [11] and Ambarisha and

2

Parker [12] used this tool to analyze a planetary gear, showing strong correlation with an analytical

lumped parameter model.

Experimental work on planetary gear dynamics has not kept pace with theoretical research.

Some experiments are conducted under static or quasi-static conditions [13]. Schlegel and Mard

[14], Toda and Botman [15], and Platt and Leopold [16] all investigated the effects of planet spac-

ing, planet position error, or tooth numbers on vibration amplitudes. They showed that the dynamic

response is particularly sensitive to these factors, which impact the symmetry of the system. The

most extensive studies were published in a series of papers by Hidaka et al. [17–20], who showed

that loads may not be distributed equally among the planets at high speed, even when they are

balanced at quasi-static speeds [17].

This research independently measures the dynamic response of each planetary gear. Exper-

iments with this level of detail have not been published in the literature, but they are necessary

to validate computer models and provide practical industry design guidance. The experimental

results presented in this paper will increase confidence in–or expose weaknesses of–current simu-

lation tools and serve as a reference for future modeling and design efforts.

EXPERIMENTAL SETUP

Two types of experiments are performed to characterize the dynamic behavior of a test planetary

gear. Modal tests apply a static torque while a modal hammer or vibration shaker excites the loaded

system. Custom-made fixtures constrain one of the planetary shafts with a load cell while static

torque is applied to the other shaft. The system frequency response is calculated from the measured

input and output signals. Spinning tests are performed under representative operating conditions.

A drive motor rotates the gears through their operating speed range while a dynamometer applies

load. Torque is held constant while the speed is gradually increased or decreased. Lubrication

temperature is also held constant.

3

Test Stand

Figure 1 shows the test stand in the modal testing configuration. A load cell at the end of a radial

arm (A) fixes the sun gear shaft to the bedplate. An excitation arm (B) is connected to the carrier

shaft where a vibration shaker (shown in photograph) or modal hammer excites the system. A

compliant coupling (C) isolates the system from the torque actuator (D).

(a)

AB

CD

Planetary Gear

(b)

Figure 1: PLANETARY GEAR MODAL TESTING SETUP (a) PHOTOGRAPH AND (b)

SCHEMATIC SHOWING (A) LOAD CELL ARM FIXING INPUT SHAFT TO BEDPLATE,

(B) EXCITATION ARM, (C) COMPLIANT COUPLING TO ISOLATE GEARBOX FROM (D)

TORQUE ACTUATOR.

Figure 2 shows the test stand in the spinning configuration. A straightforward direct drive

system eliminates signal contamination that may occur from the reaction gearbox in a power re-

circulation setup. A 250 hp (186 kW ) motor not shown at (A) drives the system. Grooved V-belts

(B) offset the gearbox from the axis of the drive motor and dynamometer to allow room for slip

rings (C) on the input and output shafts to collect data from spinning instrumentation. Using the

belts in this manner has an added benefit: they serve as a mechanical filter, isolating the test article

from the drive motor and dynamometer. Custom pulleys (D) are attached to the input/output shafts.

Four spherical thrust roller bearings (E) support each shaft. The test gearset (F) is contained inside

a housing (G). Planet slip rings (H) used to transmit data from transducers mounted directly to

the planet gear are contained inside a custom carrier. An optional compliant coupling (J) is not

4

used in these experiments1. Precise location of the planetary gear components is necessary in the

absence of system compliance from flexible or floating supports, but the more constrained system

may be preferred if necessary precision exists [3]. The 150 hp (112 kW ) load dynamometer at (K)

provides load. The symmetric nature of the test fixtures permits inversion of the test gear to drive

the sun gear or carrier. The ring gear is grounded to the gearbox housing in either configuration.

Planetary Gear

(a)

B

C D E F G H

A

J

K

(b)

Figure 2: PLANETARY GEAR SPINNING TEST CONFIGURATION (a) PHOTOGRAPH AND

(b) SCHEMATIC DIAGRAM OF PLANETARY GEAR TEST RIG. (A) INPUT FROM DRIVE

MOTOR, (B) POWER TRANSMISSION BELTS, (C) END-OF-SHAFT SLIP RINGS, (D) PUL-

LEYS, (E) SPHERICAL THRUST ROLLER BEARINGS (FOUR PER SHAFT), (F) PLANE-

TARY TEST ARTICLE, (G) GEARBOX HOUSING/GUARDS, (H) PLANET SLIP RING, (J)

OPTIONAL COMPLIANT COUPLING, (K) OUTPUT TO LOAD DYNAMOMETER.

1The coupling adds compliance to compensate for manufacturing errors, but the weight of cantilevered components

is too great to permit its use.

5

Test Article

Table 1 gives the basic dimensions of the planetary gear test article. Spur gears restrict motion to

two dimensions. A finite element program [10] shows that root stresses and static tooth deflections

are similar to practical gear operating levels for the torques used in the experiments.

Measuring the independent motion of all gear bodies is a primary objective in the experiments.

Several carrier features provide instrumentation access. Carrier components were carefully ma-

chined to minimize errors in planet pin locations that may disrupt load sharing [21]. Accelerometer

adapters were manufactured and bolted directly to the sun and planet gears.

Table 1: BASIC PLANETARY GEAR DIMENSIONS.

Sun Planet Ring

gear gears (5) gear

Number of teeth 32 41 118

Pressure angle (deg) 25 17.07

Center distance (mm) 89.66

Diametral pitch (mm) 262.64

Diameters (mm)

Base 71.25 91.29 262.71

Root 72.92 92.56 285.32

Tip 84.48 104.60 274.80

6

Instrumentation

Rotational and translational vibrations of rotating bodies (carrier, sun gear, and two of five planets)

are measured using four tangentially-mounted accelerometers. The sensors are rigidly connected

to the gear bodies near the meshing teeth to ensure measurement reliability. Figure 3 shows the

tangentially-mounted carrier accelerometers and radially-mounted ring gear accelerometers. Ring

accelerometers are used to detect radial elastic deformation as observed experimentally by Hidaka

et al. [18] and theoretically analyzed by Wu and Parker [22], although the ring gear is bolted to a

large fixed housing with sixteen bolts to achieve a nearly-rigid connection, which is assumed by

many mathematical models [5,6]. Figure 4 shows the accelerometers mounted to the planet gears.

This is an innovative aspect of these experiments, as no previously published studies have directly

measured planet gear vibration.

To acquire the data from the planet gear accelerometers (Fig. 4) during spinning tests, two

slip rings are required: one to transmit data from the planet reference frame to the carrier refer-

ence frame and another from the carrier frame to ground. Figure 2(b) shows both of these slip

rings. Once transmitted to the carrier frame through (H), the planet signals travel with the carrier

accelerometer wires through the hollow shaft to the slip ring at (C).

Figure 3: ACCELEROMETERS MOUNTED TO RING GEAR AND CARRIER.

7

Figure 4: PLANET ACCELEROMETERS ATTACHED TO RIGID ADAPTER FLANGE.

Rotational acceleration can be obtained without knowing the instantaneous orientation of the

instrumented gears while the system is spinning. Translational accelerations, however, require

knowledge of the instantaneous gear position during data collection, so encoders track the position

of each instrumented gear body. Encoders are built into the two end-of-shaft slip rings shown in

Fig. 2(b) at (C), and the planet slip rings at (H) have encoders to measure planet position respect

to the carrier. The ten-channel slip ring on the sun gear shaft has a 512 ppr encoder, and the

36-channel slip ring on the carrier shaft has a 1,024 ppr encoder. The eight-channel planet slip

rings have 1,024 ppr encoders. All encoders operate in quadrature to obtain the highest degree of

precision. Each unit has a once-per-rev indexing pulse that defines the nominal orientation of the

gear body and establishes a reference with the accelerometer mounting locations.

Data Acquisition

Twenty one accelerometers and four encoders simultaneously track the motion of all planetary

gear components during spinning tests. Torque is monitored at the drive motor and dynamometer

shafts. An analog tachometer signal is measured at either the input or output shaft, although the

encoders accurately measure gear speed and mesh frequency. Plotting the overall RMS of mea-

sured data at each speed does not provide insightful data. Several mesh harmonics excite many

natural frequencies, rendering individual peaks indistinguishable. The frequency spectra through-

8

out the speed range shows low amplitude vibration away from mesh frequency harmonics, so order

tracking isolates vibrations at each harmonic, making is easier to identify natural frequencies and

understand the dynamic response.

Data processing of modal tests is straightforward. The gear is stationary; its orientation is

known and constant, so the encoders are not needed. Static torque is measured at the sun gear

shaft. Impulse tests use standard techniques to calculate frequency response from the measured

time signals. Shaker tests apply a sine sweep excitation. There is only one excitation frequency at

each sweep step, so order tracking is not necessary. The acceleration response RMS is normalized

by the applied force RMS at each frequency step.

LUMPED PARAMETER MODEL

The analytical formulation presented by Lin and Parker [6] is the foundation of the model used

to analyze the experimental system. Rotational and translational vibrations of all gear bodies, as

well as rotation of the two pulleys (D in Fig. 2(b)) connected by the sun and carrier shafts are

considered. Figure 5(a) shows the model for modal tests with load cell and excitation arm inertias

added to the pulleys and grounded by the appropriate stiffnesses. Figure 5(b) shows the lumped

parameter model for the spinning tests with the belts connecting the pulleys to ground. Figure 6

shows the details of the planetary gear model that is common to both setups. Gear vibrations are

measured with respect to the kinematically rotating carrier reference frame {eci }. The orientation

of this reference basis is given by the carrier encoder and rotates with constant carrier speed θc in

spinning tests. The gear is stationary for modal tests with constant offset angle θc = −19.8◦ with

respect to the ground basis {Ei}. The location of the N planets on the carrier reference frame is

θpn, n = 1, ...,N. For the test case under study, N = 5 and θpn = [-20, 52, 124, 196, 268]. Further

details of the planetary gear model are given in [6].

9

Ks Kc

Ip+Ilca Ip+Imsa

Kcoup

klc

Planet 1

Planet N

Is

(a)

Planet 1

Planet N

Kc KsIp

Is

Ip

kbelt kbelt

(b)

Figure 5: OVERALL LUMPED PARAMETER MODEL OF (a) MODAL TESTS WITH

THE LOAD CELL ARM GROUNDING THE SUN PULLEY AND COMPLIANT COU-

PLING GROUNDING THE CARRIER PULLEY, AND (b) SPINNING TESTS WITH BELTS

GROUNDING EACH PULLEY.

up2Ring

uu

rc

k2

x , x , xc

us

Sun

Carrier

upn

E1

E2

e 1e2 c

c

θc

pθ

s r

y , y , yc s r

yp

xp

Figure 6: LUMPED PARAMETER MODEL OF PLANETARY GEAR WITH VIBRATIONS

MEASURED ON THE CARRIER REFERENCE FRAME ROTATING WITH CONSTANT AN-

GULAR SPEED θc.

10

The analytical model gives the matrix equation of motion with the associated displacement

vector q as

Mq+Cq+[Km+Kv(t)]q = f

q = (up1, qring, qsun, qcarrier, qplanet1, ..., qplanetN ,up2)T

qi = (xi,yi,ui)

where i = ring, sun, carrier, planet1, ..., planet N

(1)

This equation neglects the skew-symmetric gyroscopic matrix containing Coriolis acceleration

terms proportional to carrier velocity and centripetal acceleration terms in the stiffness matrix. The

effect of these quantities is negligible at operating test speeds. Experimental post-processing that

neglects Coriolis and centripetal terms shows insignificant deviations from post-processing that

accounts for them. The stiffness matrix K is divided into a mean component Km and a vibrating

component Kv(t). Km contains bearing stiffness and mean mesh stiffness terms, while Kv(t) in-

cludes only the fluctuating mesh stiffness. f is the applied external load. For all experiments in

this paper, the applied preload torque at the sun gear is 150 N ·m (110 f t · lb f ) with a reaction

torque at the carrier. This was the maximum possible torque in spinning tests, which was limited

by the capabilities of the drive motor, dynamometer, and belts. For consistency, the same torque

is applied in modal tests, although experiments at higher and lower torques did not significantly

affect the dynamic response in modal testing.

The analytical model shows that symmetry conditions inherent to planetary gears lead to three

types of modes [6]: rotational, translational, and planet modes. Rotational modes are character-

ized by purely rotational motion of the central members (sun gear, carrier, and ring gear). Transla-

tional modes are defined by purely translational motion of the central members. Planet modes are

characterized by motion in the planet gears only.

11

Modal Test Modeling

The experimental setup is stationary during modal testing, so the mesh stiffness is constant and

Kv(t) = 0. Mesh stiffness is proportional to the number of teeth in contact. The contact ratio of

this planetary gear is low. Most gear meshes will be in single tooth contact, although it is difficult to

observe each mesh cycle position because of the small backlash and difficulty of visually inspecting

each mesh once the test is assembled and preload torque is applied. An estimation of each mesh

stiffness is applied based upon the stiffness profile over one mesh cycle.

The applied force is the preload torque plus the excitation force of the impact hammer or

shaker. The applied preload torque, however, causes a static deflection that the accelerometers do

not detect. A transformation is applied so that z = q−qm, where qm is the mean value of q. z is

the dynamically fluctuating component of the original displacement vector q. Equation (1) then

reduces to

Mz+Cz+Kmtz = fmt (2)

fmt is the applied excitation from the impact hammer or shaker only. The equation of motion is

linear, and frequency domain analysis is easily performed. Impulse tests with the impact hammer

striking the tip of the radial excitation arm (B in Fig. 1) cause an applied translational force and

moment which is modeled at the pulley where the excitation arm is attached. For shaker tests, a

sinusoidal rotational and/or translational input is applied to the appropriate degrees-of-freedom,

depending on the location of the shaker attachment point. The shaker was attached to the planet

gear body along the line of action with the ring gear to apply a combined force/moment in tests

presented later in this paper.

12

Spinning Test Modeling

The transformation introduced above, q = z+qm, is applied to the spinning test system. The static

case gives fpreload = Kmqm. The parametric excitation Kv(t)z on the left hand side is ignored. This

leads to the equation of motion for spinning tests as

Mz+Cz+Kmz =−Kv(t)qm (3)

The excitation on the right hand side is the product of the fluctuating mesh stiffness matrix

Kv(t) and the mean deflection qm = K−1m fpreload .

The fluctuating mesh stiffness matrix Kv(t) requires the variation in sun-planet and ring-planet

mesh stiffness for each of the five sun-planet and ring-planet mesh pairs. A finite element model

[10] gives the mesh stiffness variation for one reference sun-planet and one reference ring-planet

mesh. The mesh phasing relationships developed in [23] give the phase difference between the

reference mesh and the remaining four meshes. Figure 7 shows the mesh stiffness variations used

to construct Kv(t). Frequency domain analysis uses the complex form of the excitation -Kv(t)qm

to compare the response to different mesh frequency excitation harmonics with the experimental

data processed by order tracking.

321 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-10

-5

0

5

10

15x 106

Mesh Cycle

N/m

4

(a)

43 215

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-10

0

10

20

30

x 106

Mesh Cycle

N/m

(b)

Figure 7: FLUCTUATING PORTION OF MESH STIFFNESS OF (a) FIVE SUN-PLANET

MESHES AND (b) FIVE RING-PLANET MESHES OVER ONE MESH CYCLE WITH 150

N ·m PRELOAD TORQUE FOR EXPERIMENTAL PLANETARY GEAR INDICATING MESH

PHASE ORDER.

13

MODAL TESTING

Impact and shaker excitation are used in modal testing. Initial experiments use the load cell arm

(A) and the excitation arm (B) as shown in Fig. 1 as impact excitation points. Data from these tests

agrees well with the analytical model below 1,000 Hz, but the high-frequency modes have small

deflection at the chosen excitation point, so correlation at higher frequencies is poor. The analytical

model shows that modes below 1,000 Hz have significant strain energy in the shafts and kinetic

energy in the carrier, sun gear, and pulleys (where the impacts are applied). They have negligible

strain energy in the gear meshes and negligible kinetic energy in the planet gears. Modes above

1,000 Hz tend to have the opposite: significant strain energy in the tooth meshes, high kinetic

energy in planetary gear components, and low deflection at the load cell and excitation arms.

Modes below 1,000 Hz are called fixture modes because they are characterized by deflection in

system components: shafts, pulleys, and the entire planetary gear as a lumped quantity. Modes

above 1,000 Hz are called gear modes because they are predominantly characterized by motion

of the individual planetary gear components. Figure 8 shows the strain energy distribution among

the system components in a typical low-frequency fixture mode and a high-frequency gear mode.

Like other fixture modes, shaft deflection dominates the 396 Hz mode. Like other gear modes,

mesh deflection and planet bearing deflection dominate the 1,831 Hz mode. Both modes shown in

Figure 8 are rotational modes. Impact testing on the excitation arm (B in Fig. 1) is used to analyze

the fixture modes. Subsequent shaker tests, with excitation directly on a planet gear, reveal the

gear modes.

Impact Testing

Figure 9 shows the frequency response of carrier rotation to an impulse at the load cell arm in

the low-frequency range. The model and experiments agree well in this region of fixture modes

dominated by strain energy in the shafts and kinetic energy in the pulleys. This impact minimally

excites the high-frequency gear modes, which contain negligible deflection at the impact point.

The 178 Hz peak shown in the experiments corresponds with a translational mode in the model. It

14

Mode 8: 1,831 HzMode 8: 1,831 Hz

Shafts Sun Ring Planet Cent Mem

Stra

in E

nerg

y Mode 3: 396 Hz

Mesh Mesh Brg Brg

Figure 8: ANALYTICAL MODEL STRAIN ENERGY DISTRIBUTION IN A ROTATIONAL

FIXTURE MODE (396 Hz) AND A ROTATIONAL GEAR MODE (1,831 Hz).

is not expected to appear in carrier rotation and suggests that there is some blending of mode types.

Alternatively, this peak may correspond to a natural frequency of the torque actuator, which might

exhibit a low-frequency rotational mode with its high inertia. The torque actuator is not modeled

because its dynamics would be well below the target frequency range.

0 100 200 300 400 500 600 700 800 900 1000

0.01

0.02

0.03

0.04

Frequency (Hz)

Car

rier R

otat

ion

(g/N

)

ModelExperiments

f1f4

f3

f2

Figure 9: FREQUENCY RESPONSE OF CARRIER ROTATION TO AN IMPACT AT THE

LOAD CELL ARM: EXPERIMENTAL MEASUREMENT AND ANALYTICAL MODEL PRE-

DICTION. PRELOAD TORQUE IS 150 N ·m AT THE SUN GEAR.

15

Shaker Testing

The analytical model predicts significant decoupling between the low-frequency shaft modes and

the high-frequency gear modes. Impact testing at the radial arms (A and B in Fig. 1) provides good

agreement among the low-frequency modes, but it is more desirable to correlate the high-frequency

modes that contain significant gear mesh deflection. An MB Dynamics Modal 50 (222 N) shaker is

mounted directly to a planet gear along its line of action with the ring gear as shown in Figure 10. A

controlled speed sweep excites the planetary gear. The excitation voltage to the shaker is adjusted

to maintain a constant force amplitude at the excitation point. Greater excitation magnitudes reveal

better dynamic response, especially in system components further away from the excitation point.

An MB Dynamics SS250VCF amplifies a sinusoidal signal generated from an HP 33120A function

generator. The data acquisition computer running LabVIEW controls the incremental excitation

frequency and amplitude of the function generator signal. A proportional controller determines

the excitation voltage amplitude based upon measurements in previous frequency steps. With

this process, 22 N (5 lb f ) is the maximum excitation amplitude obtained throughout the entire

target frequency range containing the high-frequency modes characterized by significant gear mesh

deflection, namely, 1,500 to 3,500 Hz.

Figure 10: MODAL TESTING SETUP WITH 50 lb f (222 N) SHAKER ATTACHED DIRECTLY

TO A PLANT GEAR ALONG THE LINE OF ACTION WITH THE RING GEAR. ACTUAL

INPUT FORCE IS MEASURED AT THE CONTACT POINT.

16

Excellent correlation with the analytical model is obtained with these controlled experiments.

The model predicts nine natural frequencies in this range: three rotational modes, three translation

modes, and three planet modes. Translational and planet modes have a multiplicity of two. Table

2 compares all of the natural frequencies of the planetary gear set obtained from impact testing

and shaker testing2. The dynamic response measured in experiments and the eigenvectors of the

model identify the nature of the modes. The three mode types predicted by Lin and Parker [6] are

noticeable in the experiments. Modes are numbered in increasing order per the analytical model,

which coincides with the experiments except for two cases. The horizontal line separates the low-

frequency shaft modes below 1,000 Hz (impact testing) from the high-frequency gear modes above

1,000 Hz (shaker testing). Two modes at very high frequency associated with the rigid mounting

of the ring gear are not shown.

Model parameters are not tuned to match experimental natural frequencies in Tab. 2. These

results use the baseline parameters that are physically measured (masses and bearing stiffnesses)

or estimated by finite element models (moments of inertia and mesh stiffnesses). Twelve of the

fifteen modes agree within 5%. The error in modes seven and nine may trace back to the number

of teeth actually in contact at each mesh. With a low contact ratio, the model assumes single tooth

contact for each mesh pair. Figure 7, however, shows that there will be one ring-planet mesh pair

in double tooth contact and possibly one sun-planet mesh pair in double tooth contact at any point

in the mesh cycle. Inability to know which mesh pairs are in double tooth contact may lead to this

error. Subsequent spinning tests, with mesh stiffnesses averaged over a mesh cycle, show better

agreement in these modes.

2Omitting structural modes of the housing or carrier in experiments.

17

Table 2: NATURAL FREQUENCIES OF TEST GEAR: MODAL TESTING AND ANALYTI-

CAL MODEL. IMPACT TESTS AT THE EXCITATION ARM DETERMINE MODES BELOW

1,000 Hz. SHAKER TESTS (Fig. 10) DETERMINE MODES ABOVE 1,000 Hz.

Mode Type Modal Testing Model Error

1 Trans 178 Hz 176 Hz -1%

2 Rot 217 Hz 214 Hz -1%

3 Rot 411 Hz 396 Hz -4%

4 Rot 586 Hz 583 Hz -1%

5 Trans 660 Hz 682 Hz 3%

6 Rot 840 Hz 849 Hz 1%

7 Planet 2,186 Hz 1,791 Hz -18%

8 Rot 1,893 Hz 1,831 Hz -3%

9 Trans 2,228 Hz 1,914 Hz -14%

10 Planet 2,477 Hz 2,363 Hz -5%

11 Rot 2,230 Hz 2,399 Hz 8%

12 Trans 2,556 Hz 2,470 Hz -3%

13 Planet 2,667 Hz 2,704 Hz 1%

14 Rot 2,761 Hz 2,808 Hz 2%

15 Trans 2,868 Hz 2,809 Hz -2%

18

Figure 11 shows the frequency response in planet rotation during the shaker tests described.

The locations and amplitudes of resonant peaks agree well. Figures 12 and 13 show the frequency

response in sun gear and carrier translation, respectively. The X- and Y-translations are measured

on the rotating carrier reference frame {eci } (Fig. 6). Relative amplitudes of the natural frequencies

agree well among all degrees-of-freedom. The biggest difference is the location, and in some

cases amplitude, of f9. Taking into consideration all of the natural frequencies shown in these

figures among all the degrees-of-freedom (including those not shown) and their relative amplitudes,

however, the comparisons establish significant confidence in the model.

0.2

0.4

0.6

0.8

1

1500 2000 2500 3000Frequency (Hz)

0.2

0.4

0.6

0.8

1

Plan

et R

otat

ion

(g/N

)

f15

f12f11

f13f10

f15f12

f11 f13

f10

Experiments

Model

Figure 11: FREQUENCY RESPONSE OF PLANET ROTATION COMPARING EXPER-

IMENTS TO THE LUMPED PARAMETER MODEL. SHAKER EXCITATION ON THE

PLANET GEAR BODY ALONG THE LINE OF ACTON WITH THE RING GEAR. STATIC

PRELOAD TORQUE IS 150 N ·m AT THE SUN GEAR.

19

0.05

0.10

0.15

0.20

0.25

Sun

X-T

rans

(g/N

)

1500 2000 2500 3000Frequency (Hz)

0.05

0.10

0.15

0.20

0.25

f9

f12

f15

f9

f12

f15Experiments

Model

-14% -2%

(a)

Sun

Y-T

rans

(g/N

)

f12

f15

1500 2000 2500 3000Frequency (Hz)

0.05

0.10

0.15

0.20

0.25

0.05

0.10

0.15

0.20

0.25f12

f15

Experiments

Model -3%

(b)

Figure 12: FREQUENCY RESPONSE OF SUN GEAR TRANSLATION WITH SHAKER EXCI-

TATION ON THE PLANET GEAR BODY ALONG THE LINE OF ACTON WITH THE RING

GEAR, COMPARING EXPERIMENTS TO THE LUMPED PARAMETER MODEL IN THE (a)

X- AND (b) Y-COORDINATE ON THE ROTATING CARRIER REFERENCE FRAME {eci }.

0.10

0.20

0.30

f12

f15

f9

0.10

0.20

0.30

f12

f15

f9

Car

rier X

-Tra

ns (g

/N)

1500 2000 2500 3000Frequency (Hz)

Experiments

Model

-14% -2%

(a)

0.10

0.20

0.30

f9f12

f15

Car

rier Y

-Tra

ns (g

/N)

1500 2000 2500 3000Frequency (Hz)

Experiments

Model -3%

0.10

0.20

0.30

f12

f15

(b)

Figure 13: FREQUENCY RESPONSE OF CARRIER TRANSLATION WITH SHAKER EXCI-

TATION ON THE PLANET GEAR BODY ALONG THE LINE OF ACTON WITH THE RING

GEAR, COMPARING EXPERIMENTS TO THE LUMPED PARAMETER MODEL IN THE (a)

X- AND (b) Y-COORDINATE ON THE ROTATING CARRIER REFERENCE FRAME {eci }.

20

SPINNING TESTING

Data obtained from spinning tests has higher damping, so it may be difficult to differentiate neigh-

boring modes. The results, however, clearly support modal testing data and correlation with the

model. The analytical model predicts three groups of gear modes with similar natural frequen-

cies above 1,500 Hz, each containing one planet mode, one rotational mode, and one translational

mode. In this region it may be difficult to discriminate natural frequencies grouped closely to-

gether, but the location of high amplitude response agrees with the model. Data at the lower end

of the frequency range is quite clear.

The distinct rotational, translational, and planet mode types revealed naturally in the analytical

model were present in the preceding discussion on modal testing. These modes are also identified

in spinning tests. Planet modes appear at higher frequencies near other modes and, therefore, are

difficult to distinguish in spinning tests.

Rotational Modes

Rotational modes are characterized by purely rotational motion of the central members. Figure

14 shows five natural frequencies detected in rotation of the carrier and sun gear. These natural

frequencies do not appear in translational degrees-of-freedom. Order tracking separates out the

response to the first five harmonics of mesh excitation so the natural frequencies and modes can be

examined. Vibration amplitude is plotted against response frequency, so resonant peaks appear at

the natural frequencies regardless of the harmonic number. The first and second harmonic of mesh

frequency do not excite modes above 1,500 Hz due to speed limitations of the experiments.

The low-frequency shaft modes below 1,000 Hz tend to feature predominant response in one

degree-of-freedom. Experiments show that carrier rotation dominates modes 3 and 4, whereas sun

rotation dominates mode 5. The analytical model is in agreement. Table 3 compares the rotational

modes identified in spinning tests with the analytical model prediction. The natural frequencies dif-

fer from those presented in the modal testing setup because the boundary conditions have changed

(Fig. 5) and the mean mesh stiffness over one mesh cycle is used. Parameter values with uncertain

21

100 200 300 400 500 600 700

0.1

0.2

0.3

0.4

0.5

0.6

Frequency (Hz)

Rot

atio

n (g

)

Carrier 1 harmonicst

Sun 1 harmonicst

f2

f4f5

(a)

1000 1500 2000 2500 3000

0.5

1

1.5

2

2.5

3

Frequency (Hz)

Car

rier R

otat

ion

(g)

3rd harmonic4th harmonic5th harmonic

f8

f11

(b)

Figure 14: DYNAMIC RESPONSE OF PLANETARY GEAR ROTATIONAL MOTION DUR-

ING SPINNING TESTS WITH ORDER TRACKING. (a) 1st HARMONC OF CARRIER AND

SUN GEAR. (b) 3rd THROUGH 5th HARMONC OF CARRIER ROTATION. STATIC PRELOAD

TORQUE IS 150 N ·m AT THE SUN GEAR.

accuracy are allowed to vary (typically less than 20%) to match the low-frequency modes. The

lowest and highest modes predicted by the model are not confirmed in spinning tests, but mode 14

is strongly confirmed in modal testing. Experiments show some low-amplitude response near the

first mode predicted by the model, but a natural frequency in the motor and a natural frequency in

the dynamometer are found in this region, so speculation is avoided. Four of the five modes agree

below 10% error, the other at 13%.

22

Table 3: ROTATIONAL MODES OF TEST GEARSET: SPINNING TESTS AND ANALYTICAL

MODEL. STATIC PRELOAD TORQUE IS 150 N ·m AT THE SUN GEAR.

Mode Spinning Tests Model Error

1 - 113 Hz

3 283 Hz 320 Hz -13%

4 426 Hz 413 Hz -3%

5 629 Hz 641 Hz 2%

8 1,672 Hz 1,828 Hz 9%

11 2,548 Hz 2,395 Hz -6%

Translational Modes

Translational modes are characterized by translational motion of the central members. Figure

15 shows three natural frequencies detected in translation of the carrier that are predicted by the

analytical model ( f2, f6, and f9). These modes are apparent in the sun gear as well, but plotting

sun and carrier response on the same graph was not convenient because sun translation is high in

mode 6 compared to the other peaks. The second harmonic is used in Fig. 15(a) because response

amplitudes are higher than the first harmonic. This is consistent with predictions of mesh phasing,

which indicate that translational response is minimized in the first, fourth, and fifth harmonics.

Rotational modes identified above are not excited in translational degrees-of-freedom. A finite

element model of the gearbox housing shows two modes in the vicinity of the 277 Hz and 474 Hz

natural frequencies in Fig. 15(a) that the analytical model does not capture. The 1,464 Hz natural

frequency is near a few structural modes identified by a finite element model of the carrier.

Table 4 compares the translational modes identified in spinning tests with the analytical model

prediction. As with the rotational modes presented above, some parametric tuning was allowed to

match the low-frequency modes. No tuning was performed to match the high-frequency modes.

Experiments identify two peaks in the high-frequency range, as illustrated in Fig. 15(b). The 1,949

23

100 200 300 400 500 600 700 800

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Car

rier T

rans

latio

n (g

)

2 harmonicnd

f2

f6

272 Hz 474 Hz(structural)(structural)

(a)

1000 1500 2000 2500 3000

1

2

3

4

Frequency (Hz)

Car

rier T

rans

latio

n (g

)

2nd harmonic3rd harmonic4th harmonic5th harmonic

f91,464 Hz(structural)

(b)

Figure 15: DYNAMIC RESPONSE OF CARRIER TRANSLATIONAL MOTION DUR-

ING SPINNING TESTS WITH ORDER TRACKING. (a) 2nd HARMONC OF CARRIER

Y-TRANSLATION. (b) 2nd THROUGH 5th HARMONC OF CARRIER Y-TRANSLATION.

STATIC PRELOAD TORQUE IS 150 N ·m AT THE SUN GEAR.

Hz peak probably corresponds to mode 9 in the model. This interpretation assumes that the 1,464

Hz peak is a structural mode of the carrier housing and the higher harmonics of mesh frequency

do not excite the highest two natural frequencies of the model. (Mesh phasing predicts minimized

response in the fourth and fifth harmonics, which in any case, have low excitation amplitudes.) If

the 1,464 Hz peak is actually mode 9 and the 1,949 Hz peak is mode 12, the error would be 32%

and 27%, respectively.

24

Table 4: TRANSLATIONAL MODES OF TEST GEARSET: SPINNING TESTS AND ANA-

LYTICAL MODEL. STATIC PRELOAD TORQUE IS 150 N ·m AT THE SUN GEAR.

Mode Spinning Tests Model Error

2 180 Hz 174 Hz -3%

- 272 Hz structural

- 474 Hz structural

6 670 Hz 671 Hz 0%

- 1,464 Hz structural

9 1,949 Hz 1,926 Hz -1%

12 - 2,477 Hz

15 - 2,839 Hz

CONCLUSIONS

Modal testing and spinning tests conducted under operating conditions experimentally measured

the natural frequencies and dynamic response of a production planetary gear. Sensors mounted

directly to each component give the independent motions of each gear body (sun gear, carrier, and

two planets). Post-processing resolves rotational and translational vibrations. Natural frequen-

cies and dynamic response are compared to predictions of an analytical model. Without tuning the

model parameters, modal tests confirm all natural frequencies predicted in the model; twelve of fif-

teen agree within 5%. Dynamic response–including vibration amplitude–is also confirmed across

the frequency range in nearly every degree-of-freedom. Spinning tests further confirm many of the

natural frequencies that the model predicts. The existence of rotational, translational, and planet

mode types (particularly the former two) predicted in the literature is verified experimentally. Sep-

aration of low-frequency shaft modes below 1,000 Hz and high-frequency tooth deflection modes

is demonstrated in the experiments and the model.

25

ACKNOWLEDGEMENT

This research was conducted under the guidance of Prof. Robert Parker, director of the Dynamics

and Vibrations Lab in the Mechanical and Aerospace Engineering Department of The Ohio State

University. The project was funded by the Vertical Lift Consortium and the National Rotorcraft

Technology Center, Aviation and Missile Research, Development and Engineering Center under

Technology Investment Agreement W911W6-06-2-0002, entitled National Rotorcraft Technology

Center Research Program. The authors would like to acknowledge that this research and devel-

opment was accomplished with the support and guidance of the NRTC, VLC, US Army Research

Office, and Ohio State University. The views and conclusions contained in this document are those

of the authors and should not be interpreted as representing the official policies, either expressed or

implied, of the Aviation and Missile Research, Development and Engineering Center or the U.S.

Government.

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